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        <h1 class="title">向量代数与空间解析几何</h1>
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            <span>四月 11, 2021</span>
            
  <ul class="post-tags-list" itemprop="keywords"><li class="post-tags-list-item"><a class="post-tags-list-link" href="/tags/%E6%95%B0%E5%AD%A6/" rel="tag">数学</a></li></ul>


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            <h2 id="空间向量"><a href="#空间向量" class="headerlink" title="空间向量"></a>空间向量</h2><h3 id="方向角"><a href="#方向角" class="headerlink" title="方向角"></a>方向角</h3><p>$$<br>\alpha=\frac{a_x}{|a|}=\frac{a_x}{\sqrt{a_x^2+a_y^2+a_z^2}}<br>$$</p>
<p>$$<br>\beta=\frac{a_y}{|a|}=\frac{a_y}{\sqrt{a_x^2+a_y^2+a_z^2}}<br>$$</p>
<p>$$<br>\gamma=\frac{a_z}{|a|}=\frac{a_z}{\sqrt{a_x^2+a_y^2+a_z^2}}<br>$$</p>
<h3 id="四点共面"><a href="#四点共面" class="headerlink" title="四点共面"></a>四点共面</h3><img alt="公式" style="margin:auto" src="https://www.zhihu.com/equation?tex=\left|\begin{array}{cccc}x_2-x_1%20y_2-y_1%20z_2-z_1\\x_3-x_2%20y_3-y_2%20z_3-z_2\\x_4-x_3%20y_4-y_3%20z_4-z_3\\\end{array}\right|=0">

<h3 id="混合积"><a href="#混合积" class="headerlink" title="混合积"></a>混合积</h3><p>$$<br>[abc]=(a\times b)\cdot c<br>$$</p>
<h3 id="三向量共面"><a href="#三向量共面" class="headerlink" title="三向量共面"></a>三向量共面</h3><p>$$<br>[abc] = 0<br>$$</p>
<h3 id="斜四棱柱体积"><a href="#斜四棱柱体积" class="headerlink" title="斜四棱柱体积"></a>斜四棱柱体积</h3><p>$$<br>V=[abc]<br>$$</p>
<h2 id="空间平面"><a href="#空间平面" class="headerlink" title="空间平面"></a>空间平面</h2><h3 id="点法式方程"><a href="#点法式方程" class="headerlink" title="点法式方程"></a>点法式方程</h3><p>$$<br>过一点M_0(x_0,y_0,z_0)且以\vec n={A,B,C}为法向量的平面\pi的方程<br>$$</p>
<p>$$<br>A(x-x_0)+B(y-y_0)+C(z-z_0)=0<br>$$</p>
<h3 id="一般方程"><a href="#一般方程" class="headerlink" title="一般方程"></a>一般方程</h3><p>$$<br>记D=-(Ax_0+By_0+Cz_0)<br>$$</p>
<p>$$<br>A(x-x_0)+B(y-y_0)+C(z-z_0)+D=0<br>$$</p>
<h3 id="截距式方程"><a href="#截距式方程" class="headerlink" title="截距式方程"></a>截距式方程</h3><p>$$<br>记a=-\frac{D}{A},b=-\frac{D}{B},c=-\frac{D}{C},<br>$$</p>
<p>$$<br>\frac{x}{a}+\frac{x}{b}+\frac{x}{c}=0<br>$$</p>
<h3 id="二面角"><a href="#二面角" class="headerlink" title="二面角"></a>二面角</h3><p>$$<br>\cos\theta=\frac{|n_1\cdot n_2|}{|n_1||n_2|}=\frac{|A_1A_2+B_1B_2+C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}<br>$$</p>
<h3 id="点到直线的距离"><a href="#点到直线的距离" class="headerlink" title="点到直线的距离"></a>点到直线的距离</h3><p>$$<br>d=\frac{|n\cdot\overrightarrow{M_0M_1}|}{|n|}=\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}<br>$$</p>
<h3 id="平面束方程"><a href="#平面束方程" class="headerlink" title="平面束方程"></a>平面束方程</h3><p>$$<br>\lambda(A_1x+B_1y+C_1z+D_2)+\mu(A_2x+B_2y+C_2z+D_2)=0<br>$$</p>
<h2 id="空间直线"><a href="#空间直线" class="headerlink" title="空间直线"></a>空间直线</h2><h3 id="一般方程-1"><a href="#一般方程-1" class="headerlink" title="一般方程"></a>一般方程</h3><img alt="公式" style="margin:auto" src="https://www.zhihu.com/equation?tex=\begin{equation}\left\{\begin{array}{}A_1x%2bB_1y%2bC_1z%2bD_2=0\\A_2x%2bB_2y%2bC_2z%2bD_2=0 \\\end{array}\right.\end{equation}">

<h3 id="点方向式方程-amp-对称式方程"><a href="#点方向式方程-amp-对称式方程" class="headerlink" title="点方向式方程&amp;对称式方程"></a>点方向式方程&amp;对称式方程</h3><p>$$<br>过M_0(x_0,y_0,z_0)且方向向量为s={m,n,p}的直线<br>$$</p>
<p>$$<br>\frac{x-x_0}{m}=\frac{y-y_0}{n}=\frac{z-z_0}{p}<br>$$</p>
<h3 id="参数方程"><a href="#参数方程" class="headerlink" title="参数方程"></a>参数方程</h3><img alt="公式" style="margin:auto" src="https://www.zhihu.com/equation?tex=\begin{equation}\left\{\begin{array}{}x=mt%2bx_0\\y=nt%2by_0\\z=pt%2bz_0\end{array}\right.\end{equation}">

<h3 id="线线角"><a href="#线线角" class="headerlink" title="线线角"></a>线线角</h3><p>$$<br>\cos\varphi=\frac{|s_1\cdot s_2|}{|s_1||s_2|}=\frac{|m_1m_2+n_1n_2+p_1p_2|}{\sqrt{m_1^2+n_1^2+p_1^2}\sqrt{m_2^2+n_2^2+p_2^2}}<br>$$</p>
<h3 id="线面角"><a href="#线面角" class="headerlink" title="线面角"></a>线面角</h3><p>$$<br>\sin\varphi=\frac{|s\cdot n|}{|s||n|}=\frac{|Am+Bn+Cp|}{\sqrt{m^2+n^2+p^2}\sqrt{A^2+B^2+C^2}}<br>$$</p>
<h2 id="空间曲面"><a href="#空间曲面" class="headerlink" title="空间曲面"></a>空间曲面</h2><h3 id="柱面"><a href="#柱面" class="headerlink" title="柱面"></a>柱面</h3><img alt="公式" style="margin:auto" src="https://www.zhihu.com/equation?tex=例如以平行于z轴的动直线为母线，以\Gamma为准线的柱面\\其中\Gamma\begin{equation}\left\{\begin{array}{}f(x,y)=0\\z=0\\\end{array}\right.\end{equation}\\柱面方程：f(x,y)=0">

<h3 id="旋转曲面"><a href="#旋转曲面" class="headerlink" title="旋转曲面"></a>旋转曲面</h3><p>$$<br>例如以z轴为轴，f(y,z)=0为母线的旋转曲面<br>$$</p>
<p>$$<br>令y=\pm\sqrt{x^2+y^2}<br>$$</p>
<p>$$<br>曲面方程：f(\pm\sqrt{x^2+y^2},z)=0<br>$$</p>
<h3 id="二次曲面"><a href="#二次曲面" class="headerlink" title="二次曲面"></a>二次曲面</h3><h4 id="椭球面"><a href="#椭球面" class="headerlink" title="椭球面"></a>椭球面</h4><img src="https://tc.skyone.host/blog/post/%E5%90%91%E9%87%8F%E4%BB%A3%E6%95%B0%E4%B8%8E%E7%A9%BA%E9%97%B4%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E6%A4%AD%E7%90%83%E9%9D%A2.png" alt="椭球面" style="zoom: 33%;" />
$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{y^2}{c^2}=1，(a>0,b>0,c>0)
$$

<h4 id="椭圆抛物面"><a href="#椭圆抛物面" class="headerlink" title="椭圆抛物面"></a>椭圆抛物面</h4><img src="https://tc.skyone.host/blog/post/%E5%90%91%E9%87%8F%E4%BB%A3%E6%95%B0%E4%B8%8E%E7%A9%BA%E9%97%B4%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E6%A4%AD%E7%90%83%E9%9D%A2.png" alt="椭圆抛物面" style="zoom:33%;" />
$$
\frac{x^2}{2p}+\frac{y^2}{2q}=z，(p,q同号)
$$

<h4 id="双曲抛物面"><a href="#双曲抛物面" class="headerlink" title="双曲抛物面"></a>双曲抛物面</h4><img src="https://tc.skyone.host/blog/post/%E5%90%91%E9%87%8F%E4%BB%A3%E6%95%B0%E4%B8%8E%E7%A9%BA%E9%97%B4%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%8F%8C%E6%9B%B2%E6%8A%9B%E7%89%A9%E9%9D%A2.png" alt="双曲抛物面" style="zoom:33%;" />
$$
-\frac{x^2}{2p}+\frac{y^2}{2q}=z，(p,q同号)
$$

<h4 id="单叶双曲线"><a href="#单叶双曲线" class="headerlink" title="单叶双曲线"></a>单叶双曲线</h4><img src="https://tc.skyone.host/blog/post/%E5%90%91%E9%87%8F%E4%BB%A3%E6%95%B0%E4%B8%8E%E7%A9%BA%E9%97%B4%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%8D%95%E5%8F%B6%E5%8F%8C%E6%9B%B2%E7%BA%BF.png" alt="单叶双曲线" style="zoom: 40%;" />
$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{y^2}{c^2}=1，(a>0,b>0,c>0)
$$

<h4 id="双叶双曲线"><a href="#双叶双曲线" class="headerlink" title="双叶双曲线"></a>双叶双曲线</h4><img src="https://tc.skyone.host/blog/post/%E5%90%91%E9%87%8F%E4%BB%A3%E6%95%B0%E4%B8%8E%E7%A9%BA%E9%97%B4%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%8F%8C%E5%8F%B6%E5%8F%8C%E6%9B%B2%E7%BA%BF.png" alt="双叶双曲线" style="zoom:40%;" />
$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{y^2}{c^2}=-1，(a>0,b>0,c>0)
$$
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			<ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%A9%BA%E9%97%B4%E5%90%91%E9%87%8F"><span class="toc-number">1.</span> <span class="toc-text">空间向量</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%96%B9%E5%90%91%E8%A7%92"><span class="toc-number">1.1.</span> <span class="toc-text">方向角</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%9B%9B%E7%82%B9%E5%85%B1%E9%9D%A2"><span class="toc-number">1.2.</span> <span class="toc-text">四点共面</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%B7%B7%E5%90%88%E7%A7%AF"><span class="toc-number">1.3.</span> <span class="toc-text">混合积</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%B8%89%E5%90%91%E9%87%8F%E5%85%B1%E9%9D%A2"><span class="toc-number">1.4.</span> <span class="toc-text">三向量共面</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%96%9C%E5%9B%9B%E6%A3%B1%E6%9F%B1%E4%BD%93%E7%A7%AF"><span class="toc-number">1.5.</span> <span class="toc-text">斜四棱柱体积</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%A9%BA%E9%97%B4%E5%B9%B3%E9%9D%A2"><span class="toc-number">2.</span> <span class="toc-text">空间平面</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%82%B9%E6%B3%95%E5%BC%8F%E6%96%B9%E7%A8%8B"><span class="toc-number">2.1.</span> <span class="toc-text">点法式方程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%B8%80%E8%88%AC%E6%96%B9%E7%A8%8B"><span class="toc-number">2.2.</span> <span class="toc-text">一般方程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%88%AA%E8%B7%9D%E5%BC%8F%E6%96%B9%E7%A8%8B"><span class="toc-number">2.3.</span> <span class="toc-text">截距式方程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%BA%8C%E9%9D%A2%E8%A7%92"><span class="toc-number">2.4.</span> <span class="toc-text">二面角</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%82%B9%E5%88%B0%E7%9B%B4%E7%BA%BF%E7%9A%84%E8%B7%9D%E7%A6%BB"><span class="toc-number">2.5.</span> <span class="toc-text">点到直线的距离</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%B9%B3%E9%9D%A2%E6%9D%9F%E6%96%B9%E7%A8%8B"><span class="toc-number">2.6.</span> <span class="toc-text">平面束方程</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%A9%BA%E9%97%B4%E7%9B%B4%E7%BA%BF"><span class="toc-number">3.</span> <span class="toc-text">空间直线</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%B8%80%E8%88%AC%E6%96%B9%E7%A8%8B-1"><span class="toc-number">3.1.</span> <span class="toc-text">一般方程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%82%B9%E6%96%B9%E5%90%91%E5%BC%8F%E6%96%B9%E7%A8%8B-amp-%E5%AF%B9%E7%A7%B0%E5%BC%8F%E6%96%B9%E7%A8%8B"><span class="toc-number">3.2.</span> <span class="toc-text">点方向式方程&amp;对称式方程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B"><span class="toc-number">3.3.</span> <span class="toc-text">参数方程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%BA%BF%E7%BA%BF%E8%A7%92"><span class="toc-number">3.4.</span> <span class="toc-text">线线角</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%BA%BF%E9%9D%A2%E8%A7%92"><span class="toc-number">3.5.</span> <span class="toc-text">线面角</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%A9%BA%E9%97%B4%E6%9B%B2%E9%9D%A2"><span class="toc-number">4.</span> <span class="toc-text">空间曲面</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%9F%B1%E9%9D%A2"><span class="toc-number">4.1.</span> <span class="toc-text">柱面</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%97%8B%E8%BD%AC%E6%9B%B2%E9%9D%A2"><span class="toc-number">4.2.</span> <span class="toc-text">旋转曲面</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%BA%8C%E6%AC%A1%E6%9B%B2%E9%9D%A2"><span class="toc-number">4.3.</span> <span class="toc-text">二次曲面</span></a><ol class="toc-child"><li class="toc-item toc-level-4"><a class="toc-link" href="#%E6%A4%AD%E7%90%83%E9%9D%A2"><span class="toc-number">4.3.1.</span> <span class="toc-text">椭球面</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E6%A4%AD%E5%9C%86%E6%8A%9B%E7%89%A9%E9%9D%A2"><span class="toc-number">4.3.2.</span> <span class="toc-text">椭圆抛物面</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E5%8F%8C%E6%9B%B2%E6%8A%9B%E7%89%A9%E9%9D%A2"><span class="toc-number">4.3.3.</span> <span class="toc-text">双曲抛物面</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E5%8D%95%E5%8F%B6%E5%8F%8C%E6%9B%B2%E7%BA%BF"><span class="toc-number">4.3.4.</span> <span class="toc-text">单叶双曲线</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E5%8F%8C%E5%8F%B6%E5%8F%8C%E6%9B%B2%E7%BA%BF"><span class="toc-number">4.3.5.</span> <span class="toc-text">双叶双曲线</span></a></li></ol></li></ol></li></ol>	
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